Transformations of Functions
Transformations alter a function while maintaining the original characteristics of that function.
Learning Objective

Determine whether a given transformation is an example of translation, scaling, rotation, or reflection
Key Points
 Transformations are ways that a function can be adjusted to create new functions.
 Transformations often preserve the original shape of the function.
 Common types of transformations include rotations, translations, reflections, and scaling (also known as stretching/shrinking).
Terms

translation
Shift of an entire function in a specific direction.

Scaling
Changes the size and/or the shape of the function.

rotation
Spins the function around the origin.

reflection
Mirror image of a function.
Full Text
A transformation takes a basic function and changes it slightly with predetermined methods. This change will cause the graph of the function to move, shift, or stretch, depending on the type of transformation. The four main types of transformations are translations, reflections, rotations, and scaling.
Translations
A translation moves every point by a fixed distance in the same direction. The movement is caused by the addition or subtraction of a constant from a function. As an example, let
Graph of a function being translated
The function
Reflections
A reflection of a function causes the graph to appear as a mirror image of the original function. This can be achieved by switching the sign of the input going into the function. Let the function in question be
Graph of a function being reflected
The function
Rotations
A rotation is a transformation that is performed by "spinning" the object around a fixed point known as the center of rotation. Although the concept is simple, it has the most advanced mathematical process of the transformations discussed. There are two formulas that are used:
Where
Scaling
Scaling is a transformation that changes the size and/or the shape of the graph of the function. Note that until now, none of the transformations we discussed could change the size and shape of a function  they only moved the graphical output from one set of points to another set of points. As an example, let
Graph of a function being scaled
The function
Key Term Reference
 Reflection
 Appears in these related concepts: Reflections, Symmetry of Functions, and Parabolas As Conic Sections
 constant
 Appears in these related concepts: Graphing Quadratic Equations in Vertex Form, Inverse Variation, and Direct Variation
 degree
 Appears in these related concepts: Partial Fractions, Solving Quadratic Equations by Factoring, and Adding and Subtracting Polynomials
 direction
 Appears in these related concepts: Parallel and Perpendicular Lines, Slope, and Applications of the Parabola
 distance
 Appears in these related concepts: Inequalities with Absolute Value, The Distance Formula and Midpoints of Segments, and Linear Mathematical Models
 expression
 Appears in these related concepts: Simplifying, Multiplying, and Dividing Rational Expressions, Compound Inequalities, and Sets of Numbers
 function
 Appears in these related concepts: The Vertical Line Test, Visualizing Domain and Range, and Solving Differential Equations
 graph
 Appears in these related concepts: Graphing Equations, Graphs of Equations as Graphs of Solutions, and Graphical Representations of Functions
 output
 Appears in these related concepts: A Study of Process, Introducing Aggregate Supply, and Functions and Their Notation
 point
 Appears in these related concepts: Polynomial and Rational Functions as Models, Relative Minima and Maxima, and Graphing Quadratic Equations In Standard Form
 scaling
 Appears in these related concepts: Stretching and Shrinking, Types of Conic Sections, and Introduction to Ellipses
 set
 Appears in these related concepts: Sequences, Introduction to Sequences, and Sequences of Mathematical Statements
 sign
 Appears in these related concepts: The Rule of Signs, Polynomial Inequalities, and The Intermediate Value Theorem
Sources
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